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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Fréchet space} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{frchet_spaces}{}\section*{{Fr\'e{}chet spaces}}\label{frchet_spaces} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{duals}{Duals}\dotfill \pageref*{duals} \linebreak \noindent\hyperlink{AsProjectiveLimits}{As projective limits}\dotfill \pageref*{AsProjectiveLimits} \linebreak \noindent\hyperlink{path_smoothness}{Path smoothness}\dotfill \pageref*{path_smoothness} \linebreak \noindent\hyperlink{DifferentiableAndSmoothFunctions}{Differentiable and smooth functions}\dotfill \pageref*{DifferentiableAndSmoothFunctions} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Fr\'e{}chet spaces are particularly well-behaved [[topological vector spaces]] (TVSes). Every [[Cartesian space]] $\mathbb{R}^n$ is a Fr\'e{}chet space, but Fr\'e{}chet spaces may have non-[[finite number|finite]] [[dimension]]. There is [[analysis]] on Fr\'e{}chet spaces, yet they are more general than [[Banach spaces]]; as such, they are popular as local model spaces for possibly [[infinite-dimensional manifolds]]: \emph{[[Fréchet manifolds]]}. A basic example of a Fr\'e{}chet space is $\mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim} \mathbb{R}^n$, as a [[topological space]] the \emph{[[projective limit]]} over the finite dimensional [[Cartesian spaces]] $\mathbb{R}^n$ (example \ref{ProjRInfinity} below) . This is not a [[Banach space]] anymore, since it does not carry a compatible [[norm]] anymore (e.g. \hyperlink{Saunders89}{Saunders 89}, p. 253). But it evidently does carry the functions $\mathbb{R}^\infty \overset{p^n}{\longrightarrow} \mathbb{R}^n \overset{\Vert -\Vert_n}{\longrightarrow} \mathbb{R}$ for all $n \in \mathbb{N}$, where $p^n$ is the defining [[projection]] and where ${\Vert -\Vert}_n$ is the standard [[norm]] on $\mathbb{R}^n$. While not norms, these composites are [[seminorms]] on $\mathbb{R}^\infty$, they only fail the non-degeneracy condition saying that only the 0-vector has vanishing norm. Generallly, a Fr\'e{}chet space is equivalently a [[real vector space]] equipped with a countable familiy of [[seminorms]], with compatibility conditions modeled on this example. See def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms} below. Beware the clash ofterminology: a `Fr\'e{}chet topology' on a `Fr\'e{}chet topological space' is something different; this just means that a [[topological space]] satisfies the $T_1$ [[separation axiom]]. (Like all [[Hausdorff space|Hausdorff]] [[topological vector spaces]], Fr\'e{}chet spaces satisfy this axiom, but they have a good deal of additional [[structure]] and [[properties]].) \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} There are various equivalent definitions of Fr\'e{}chet spaces: \begin{defn} \label{FrechetSpaceAsMetisableCompleteTVS}\hypertarget{FrechetSpaceAsMetisableCompleteTVS}{} A \textbf{Fr\'e{}chet space} is equivalently a [[complete space|complete]] [[Hausdorff space|Hausdorff]] [[locally convex vector space]] that is [[metrisable topological space|metrisable]]. The metric can be chosen to be translation-invariant. \end{defn} \begin{defn} \label{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}\hypertarget{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}{} \textbf{(via systems of [[seminorms]])} A \textbf{Fr\'e{}chet space} is a [[complete space|complete]] [[Hausdorff space|Hausdorff]] [[topological vector space]] $V$ whose [[topological space|topology]] may be given (as a [[gauge space]]) by a [[countable set|countable]] family of [[seminormed vector space|seminorms]], hence for which there exists a family of [[seminorms]] \begin{displaymath} {\Vert - \Vert}_n \;\colon\; V \longrightarrow \mathbb{R} \,, \;\;\; n \in \mathbb{N} \end{displaymath} such that the set of all [[open balls]] of the form \begin{displaymath} B_\epsilon^{(n)}(x) \coloneqq \left\{ y \in V \,\vert\, {\Vert x-y \Vert}_n \lt \epsilon \right\} \;\;\;\;\;\;\;\; for \; x \in V\,, \epsilon \gt 0 \,, n \in \mathbb{N} \end{displaymath} is a [[base of neighborhoods]] of $x$. \end{defn} \begin{defn} \label{}\hypertarget{}{} We accept as an [[automorphism]] of Fr\'e{}chet spaces any [[linear map|linear]] [[homeomorphism]]; in particular, the particular translation-invariant metric or countable family of [[seminorms]] used to prove that a space is a Fr\'e{}chet space is \emph{not} required to be preserved. More generally, the [[morphisms]] of Fr\'e{}chet spaces are the [[continuous map|continuous]] linear maps, so that Fr\'e{}chet spaces form a [[full subcategory]] of the category $TVS$ of [[topological vector spaces]]. \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{}\hypertarget{}{} Every [[Banach space]] is a Fr\'e{}chet space. \end{example} \begin{example} \label{}\hypertarget{}{} If $X$ is a [[compact space|compact]] [[smooth manifold]], then the space of [[smooth map|smooth maps]] on $X$ is a Fr\'e{}chet space. This can be extended to some non-compact manifolds, in particular when $X$ is the [[real line]]. \end{example} \begin{example} \label{SchwartSpaceOfFunctionsIsFrechetSpace}\hypertarget{SchwartSpaceOfFunctionsIsFrechetSpace}{} \textbf{(the [[Schwartz space]] is a Fr\'e{}chet space)} The [[Schwartz space]] $\mathcal{S}$ of functions with rapidly decreasing derivatives (\href{Schwartz+space#SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}{this def.}) is a Fr\'e{}chet space. \end{example} See \href{Schwartz+space#SchwartSpaceOfFunctionsIsFrechetSpace}{this prop.} \begin{example} \label{}\hypertarget{}{} The [[Lebesgue space]] $L^p(\mathbb{R})$ for $p \lt 1$ is \emph{not} a Fr\'e{}chet space, because it is not locally convex. \end{example} \begin{example} \label{ProjRInfinity}\hypertarget{ProjRInfinity}{} Consider the [[direct product]] (as [[topological vector spaces]]) of a countable number of copies of the [[real line]] $\mathbb{R}$ Equivalently the [[projective limit]] (as [[topological vector spaces]]) \begin{displaymath} \mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{R}^n = \underset{\longleftarrow}{\lim} \left( \cdots \to \mathbb{R}^2 \overset{}{\to} \mathbb{R}^1 \to \mathbb{R}^0 \right) \end{displaymath} over all [[Cartesian spaces]] via their canonical [[projection maps]]. (Beware that the same symbol ``$\mathbb{R}^\infty$'' is also used for the limit of the same sequence but with $\mathbb{R}^n$ with discrete topology, what leads to a [[linearly compact vector space]] as well as for the [[direct sum]]/[[inductive limit]] of $\mathbb{R}\to \mathbb{R}^2\hookrightarrow\mathbb{R}^3\hookrightarrow\ldots$, which is different.) Write \begin{displaymath} \pi_n \colon \mathbb{R}^\infty \longrightarrow \mathbb{R}^n \end{displaymath} for the induced [[projection]] maps onto the first $n$ copies and let $\|\cdot\|_n$ be the canonical [[norm]] on $\mathbb{R}^n$. Then a compatible countable family of seminorms on $\mathbb{R}^\infty$, according to def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}, is given by $v \mapsto {\Vert\pi_n(v) \Vert_n}$. Hence equipped with these, $\mathbb{R}^\infty$ becomes a Fr\'e{}chet space. \end{example} On the other hand, the locally convex direct sum of a countable number of copies of $\mathbb{R}$ is not a Fr\'e{}chet space. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Fr\'e{}chet spaces are [[barrelled]] and [[bornological]]. \hypertarget{duals}{}\subsubsection*{{Duals}}\label{duals} The [[dual space|dual]] [[topological vector space]] of a Fr\'e{}chet space $F$ is itself again a Fr\'e{}chet space precisely only if $F$ is in fact a [[Banach space]]. This follows from the statement paragraph 29.1 (7) in (\hyperlink{Koethe}{Koethe}), which is: The strong dual of a locally convex metrizable TVS $F$ is metrizable iff $F$ is normable. See also (\hyperlink{Saunders89}{Saunders 89, p. 255}). \hypertarget{AsProjectiveLimits}{}\subsubsection*{{As projective limits}}\label{AsProjectiveLimits} Every [[complete topological space|complete]] [[locally convex topological vector space]] $X$ is the [[cofiltered category|cofiltered]] [[projective limit]] of [[Banach spaces]] in the [[category]] of [[locally convex spaces]]. (Note that Fr\'e{}chet spaces are additionally required to be \emph{metrisable}, so this is more general.) To see this, choose a base $\{U_{\alpha}\}_{\alpha \in A}$ of the neighborhood filter of $0$, consisting of convex, balanced and absorbing sets and let $p_{\alpha}$ be Minkowski functional associated to $U_{\alpha}$. The Hausdorffification $X_{\alpha}$ of $(X, p_{\alpha})$ is easily seen to be a Banach space and because $A$ is directed by reverse inclusion so is $X_{\alpha}$. It is straightforward to check that $X = \underset{\longleftarrow}{\lim} X_{\alpha}$ in the category of locally convex spaces. For details, see (\hyperlink{SchaeferWolff99}{Schaefer-Wolff 99, Chapter~II.\S{}5}, \href{http://books.google.com/books?id=9kXY742pABoC&pg=PA51}{page~51ff}. Now given that a Fr\'e{}chet space admits a \emph{decreasing sequence} of convex balanced and absorbing neighborhoods, it follows immediately that: Every Fr\'e{}chet space is a [[sequential limit|sequential]] [[projective limit]] of [[Banach spaces]]. Conversely, any limit of a countable sequence of Banach spaces is a Fr\'e{}chet space. See (\hyperlink{SchaeferWolff99}{Schaefer-Wolff 99}), Chapter II.\S{}4, page 48f (as well as Theorem I.6.1, page 28). (from \href{http://math.stackexchange.com/a/53020/58526}{this math.stackexchange comment}) See also example \ref{ProjRInfinity} below. \hypertarget{path_smoothness}{}\subsubsection*{{Path smoothness}}\label{path_smoothness} \begin{defn} \label{PathSmoothFunction}\hypertarget{PathSmoothFunction}{} \textbf{(path smooth linear function)} Let $V$ be a Fr\'e{}chet vector space (def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}). Then a [[linear function]] \begin{displaymath} \mu \;\colon\; V \longrightarrow \mathbb{R} \end{displaymath} is called \emph{path smooth} if for every [[smooth function]] \begin{displaymath} g \;\colon\; \mathbb{R} \longrightarrow V \end{displaymath} the [[composition|composite]] \begin{displaymath} \mu \circ g \;\colon\; \mathbb{R} \longrightarrow \mathbb{R} \end{displaymath} is a [[smooth function]]. \end{defn} \begin{defn} \label{PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous}\hypertarget{PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous}{} \textbf{(path-smooth linear functions on a Fr\'e{}chet space are continuous)} Let $V$ be a Fr\'e{}chet vector space (def. \ref{FrechetSpaceAsCompleteTVSWithCompatibleSeminorms}). Then every [[linear function]] on $V$ which is path-smooth (def. \ref{PathSmoothFunction}) is [[continuous function|continuous]]. \end{defn} (\hyperlink{MoerdijkReyes91}{Moerdijk-Reyes 91, chapter II, lemma 3.7}) Prop. \ref{PathSmoothLinearFunctionsOnFrechetSpaceAreContinuous} implies for instance that \emph{[[distributions are the smooth linear functionals]]}. See there for more. \hypertarget{DifferentiableAndSmoothFunctions}{}\subsection*{{Differentiable and smooth functions}}\label{DifferentiableAndSmoothFunctions} It is possible to generalize some aspects of [[analysis]] (differential calculus) to Fr\'e{}chet spaces (e.g. \hyperlink{Michor80}{Michor 80, chapter 8}, \hyperlink{Saunders89}{Saunders 89, p. 256}). For example the definition of the [[derivative]] of a curve is simply the same as in finite dimensions: \begin{defn} \label{}\hypertarget{}{} For a continuous path in a Fr\'e{}chet space $f(t)$ we define \begin{displaymath} f'(t) = \lim_{h \to 0} \frac{1}{h} (f(t + h) - f(t)) \end{displaymath} If the limit exists and is continuous, we say that $f$ is continuously differentiably or $C^1$. \end{defn} And just as in the finite dimensional case, we can define the partial derivative, or rather: the [[directional derivative|directional]] or G\^a{}teaux derivative: \begin{defn} \label{}\hypertarget{}{} \textbf{directional derivative} Let $F$ and $G$ be Fr\'e{}chet spaces, $U \subseteq F$ open and $P: U \to G$ a nonlinear continuous map. The derivative of $P$ at the point $f \in U$ in the direction $h \in F$ is the map \begin{displaymath} D P: U \times F \to G \end{displaymath} \begin{displaymath} D P(f) h = \lim_{t \to 0} \frac{1}{t} ( P(f + t h) - P(f)) \end{displaymath} If the limit exists and is jointly continuous in both variables we say that $P$ is continuous differentiable or $C^1$. \end{defn} A simple, but nontrivial example is the operator \begin{displaymath} P: C^{\infty}[a, b] \to C^{\infty}[a, b] \end{displaymath} \begin{displaymath} P(f) \coloneqq f f' \end{displaymath} with the derivative \begin{displaymath} D P(f) h = f'h + f h' \end{displaymath} It is possible to generalize the Riemann integral to Fr\'e{}chet spaces, too: For a continuous path $f(t)$ on an interval $[a, b]$ in a Fr\'e{}chet space $F$ we look for an element $\int_a^b f(t) d t \in F$. It turns out that such an element exists and is unique, if we impose some properties of the integral known from the finite dimensional case: \begin{theorem} \label{}\hypertarget{}{} There exists a unique element $\int_a^b f(t) d t \in F$ such that (i) for every continuous functional $\phi$ we have $\phi(\int_a^b f(t) d t) = \int_a^b \phi(f(t)) d t$, (ii) for every continuous seminorm ${\| \cdot \|}$ we have ${\| \int_a^b f(t) d t \|} \leq \int_a^b {\| f(t) \|} d t$ (iii) integration is linear and (iv) additive, i.e. $\int_a^b f(t) d t + \int_b^c f(t) d t = \int_a^c f(t) d t$ \end{theorem} There is a version of the fundamental theorem of calculus: \begin{theorem} \label{}\hypertarget{}{} If P is $C^1$ and $f + t h \in Domain(P)$ for $0 \leq t \leq 1$, then \begin{displaymath} P(f + h) - P(f) = \int_0^1 D P(f + t h) \;h \; d t \end{displaymath} \end{theorem} The [[chain rule]] is valid: \begin{theorem} \label{}\hypertarget{}{} If $P$ and $Q$ are $C^1$ then so is their composition $Q \circ P$ and \begin{displaymath} D [Q \circ P](f) h = D Q(P(f)) \; D P(f) \; h \end{displaymath} \end{theorem} The first derivative $D P$ is a function of two variables, the base point $f$ and the direction $h$. Since $D P$ is already linear in $h$, we define the second derivative with respect to $f$ only: \begin{defn} \label{}\hypertarget{}{} \textbf{second derivative} The second derivative of $P$ in the direction $k$ is defined to be \begin{displaymath} D^2 P(f) (h, k) = \lim_{t \to 0} \frac{1}{t} (D P(f + t k) h - D P(f) h) \end{displaymath} \end{defn} It is a theorem that the second derivative, if it exists and is jointly continuous, is bilinear in $(h ,k)$. We can iterate this procedure to define derivatives of arbitrary order, and thus the notion of \textbf{smooth functions between Fr\'e{}chet spaces}. This allows to define the concept of smooth \emph{[[Fréchet manifolds]]}. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Dietmar Vogt]], \emph{Lectures on Fr\'e{}chet spaces}, 2000 (\href{http://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf}{pdf}) \item PlanetMath, \emph{\href{http://planetmath.org/frechetspace}{Fr\'e{}chet space}} \item Wikipedia, \emph{\href{https://secure.wikimedia.org/wikipedia/en/wiki/Fréchet_space}{Fr\'e{}chet space}} \item Gottfried Koethe: \emph{Topological Vector Spaces I} \item H. Schaefer, Manfred Wolff \emph{\href{http://books.google.com/books?id=9kXY742pABoC}{Topological Vector Spaces}}, Springer (1999) \item [[Andreas Kriegl]], \emph{Fr\'e{}chet Spaces} lecture notes, 2016 (\href{http://www.mat.univie.ac.at/~kriegl/Skripten/2016SS.pdf}{pdf}) \end{itemize} Discussion of analysis on Fr\'e{}chet spaces includes \begin{itemize}% \item [[Peter Michor]], chapter 8 of \emph{Manifolds of differentiable mappings}, Shiva Publishing (1980) \href{http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf}{pdf} \item Richard S. Hamilton: \emph{The Inverse Function Theorem of Nash and Moser} (Bulletin (New Series) of the American Mathematical Society Volume 7, Number 1, July 1982) \end{itemize} Discussion in the context of [[jet bundles]] and [[locally pro-manifolds]] includes \begin{itemize}% \item [[David Saunders]], chapter 7 of \emph{The geometry of jet bundles}, London Mathematical Society Lecture Note Series \textbf{142}, Cambridge Univ. Press 1989. \item [[Igor Khavkine]], [[Urs Schreiber]], section 2.2 of \emph{[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]} (\href{https://arxiv.org/abs/1701.06238}{arXiv:1701.06238}) \end{itemize} Discussion in the context of [[synthetic differential geometry]] includes \begin{itemize}% \item [[Ieke Moerdijk]], [[Gonzalo Reyes]], around lemma 3.7 in chapter II of \emph{[[Models for Smooth Infinitesimal Analysis]]}, Springer 1991 \end{itemize} Refinement to [[noncommutative geometry]] by suitable smoothed [[C-star-algebras]] is discussed in \begin{itemize}% \item Nikolay Ivankov, \emph{Unbounded bivariant K-theory and an Approach to Noncommutative Fr\'e{}chet spaces} \href{http://hss.ulb.uni-bonn.de/2011/2624/2624.pdf}{pdf} \end{itemize} [[!redirects Frechet space]] [[!redirects Frechet spaces]] [[!redirects Fréchet space]] [[!redirects Fréchet spaces]] [[!redirects Fréchet vector space]] [[!redirects Fréchet vector spaces]] [[!redirects Frechet vector space]] [[!redirects Frechet vector spaces]] \end{document}